Celebratio Mathematica

The let­ters which are dis­played here provide a breath­tak­ing di­ary of In­grid Daubech­ies’ sci­entif­ic achieve­ments on wave­lets.

The wave­let re­volu­tion blew up in the early eighties. It cre­ated an im­mense en­thu­si­asm and an un­usu­al un­der­stand­ing between sci­ent­ists work­ing in com­pletely dis­tinct fields. We be­lieved that bar­ri­ers in com­mu­nic­a­tion would dis­ap­pear for ever. We thought we were build­ing a united sci­ence. How did this start? It began in the six­ties. Dav­id Hu­bel and Tor­sten Wies­el had dis­covered that some proto-wave­lets were sem­in­al in the func­tion­ing of the primary visu­al cor­tex of mam­mals. In his fam­ous book (Vis­ion, MIT Press, 1982) Dav­id Marr pro­posed a fas­cin­at­ing al­gorithm which could emu­late this pro­cessing. Jacques Magnen, Ro­land Sénéor, and Ken­neth Wilson, were us­ing some oth­er proto-wave­lets in quantum field the­ory. Phase space loc­al­iz­a­tion was the fa­vor­ite tool of James Glimm and Ar­thur Jaffe. Un­fold­ing a sig­nal in the time fre­quency plane was pro­posed by Eu­gene Wign­er in the thirties. Wign­er’s mo­tiv­a­tion was quantum mech­an­ics. Wign­er was fol­lowed by Den­nis Gabor and Claude Shan­non in the forties. In math­em­at­ics Al­berto Calderón had already in­tro­duced a vari­ant of a wave­let ana­lys­is as an al­tern­at­ive to Four­i­er ana­lys­is. In the late sev­en­ties wave­lets were pop­ping up every­where. But the needed uni­fic­a­tion was still miss­ing un­til the French geo­phys­i­cist Jean Mor­let (1931–2007) made the fun­da­ment­al dis­cov­ery which is de­scribed now. Mor­let was work­ing for the Oil com­pany Elf Aquitaine (now Total) and his re­search was mo­tiv­ated by prob­lems en­countered in ana­lyz­ing some re­flec­ted seis­mic waves. These seis­mic waves were a key tool in oil pro­spect­ing. Work­ing on these spe­cif­ic sig­nals, Mor­let ex­per­i­mented with the flaws of win­dowed Four­i­er ana­lys­is. Then he in­tro­duced a re­volu­tion­ary meth­od which would later be named “wave­let ana­lys­is” and ob­served that this new tool was more ef­fi­cient than what was used be­fore. Twenty years later he was awar­ded the Re­gin­ald Fessenden Prize. In 1997 Pierre Goupil­laud in­tro­duced Jean Mor­let with these words:

Mor­let per­formed the ex­cep­tion­al feat of dis­cov­er­ing a nov­el math­em­at­ic­al tool which has made the Four­i­er trans­form ob­sol­ete after 200 years of uses and ab­uses, par­tic­u­larly in its fast ver­sion…Un­til now, his only re­ward for years of per­sever­ance and cre­ativ­ity in pro­du­cing this ex­traordin­ary tool was an early re­tire­ment from ELF.

Goupil­laud was right. Soon after his dis­cov­ery Mor­let was fired by ELF and suffered a break­down. At that time Mor­let had a vis­ion without a proof and needed some help. Alex Gross­mann (1930–2019), a phys­i­cist at the Centre de Physique Théorique, Mar­seilles-Lu­miny, listened to Mor­let with much pa­tience and sym­pathy. Fi­nally Alex un­der­stood Mor­let’s claim and gave it the sound math­em­at­ic­al for­mu­la­tion which is de­scribed now.

Let \( s(t) \) be a sig­nal with a fi­nite en­ergy. Its wave­let trans­form \( W(a, x) \) is a con­tinu­ous func­tion of two vari­ables \( a > 0 \) and \( x\in \mathbb{R} \). This wave­let trans­form de­pends on the sig­nal \( s(t) \) but also on the wave­let \( \psi \) which is used in the ana­lys­is. How much the choice of this wave­let will af­fect the res­ults is an im­port­ant prob­lem. Be­fore dis­cuss­ing it let us re­turn to the defin­i­tion of the wave­let trans­form. In this defin­i­tion \( 1/a \) is the mag­ni­fic­a­tion of the “math­em­at­ic­al mi­cro­scope” \( \psi \), and \( x \) is a real num­ber in­dic­at­ing the place where you want to zoom in­to the sig­nal \( s(t). \) Fi­nally \( W(a, x) \) is the cross-cor­rel­a­tion \( \langle s, \psi_{a, x}\rangle \) between the sig­nal \( s \) and the shrunk or dilated wave­let \[ \psi_{a, x}=\frac{1}{a}\psi\biggl(\frac{t-x}{a}\biggr). \] We have \[ W(a,x)=\int_{-\infty}^{+\infty} \overline{\psi_{a, x}}(t)s(t)\,dt. \] The func­tion \( \psi \) is named the ana­lyz­ing wave­let (the mi­cro­scope) and the over­line means com­plex con­jug­ate. This ana­lyz­ing wave­let shall be a smooth func­tion, loc­al­ized around 0, and os­cil­lat­ing. This last re­quire­ment means \( \int_{-\infty}^{+\infty} \psi(t)\,dt=0. \) Fi­nally \( \psi \) shall be an ad­miss­ible wave­let: both \[ \int_0^{\infty}|\widehat \psi(u)|^2\,\frac{du}{u}=1 \quad \text{ and } \quad \int_0^{\infty}|\widehat \psi(-u)|^2\,\frac{du}{u}=1 \] are needed. Here \( \widehat \psi \) is the Four­i­er trans­form of \( \psi \). Any func­tion \( \psi \) which is real val­ued, smooth, loc­al­ized, and os­cil­lat­ing be­comes an ad­miss­ible wave­let after mul­ti­plic­a­tion by a suit­able con­stant.

Then Gross­mann and Mor­let proved that every sig­nal \( s(t) \) can be ex­actly re­con­struc­ted by a simple in­ver­sion for­mula. (See A. Gross­mann and J. Mor­let, “De­com­pos­i­tion of Hardy func­tions in­to square in­teg­rable wave­lets of con­stant shape,” SIAM J. Math. Anal. 15 (1984) 723–736.) Everything works as if the ana­lyz­ing wave­lets \( \psi_{a, x} \) were an or­thonor­mal basis. In­deed un­der the as­sump­tion that \( \psi \) is an ad­miss­ible wave­let we have \[ s(t)=\int_0^{\infty}\!\!\!\int_{-\infty}^{+\infty}W(a, x) {\psi_{a, x}}(t)\,dx\,\frac{da}{a}. \]

As it was an­nounced by Mor­let, wave­let ana­lys­is provides us with a bet­ter un­der­stand­ing of sig­nals which can­not be ana­lyzed cor­rectly by a stand­ard win­dowed Four­i­er ana­lys­is (Gabor wave­lets, 1946). This is the case when strong tran­si­ents oc­cur in the sig­nal. It is also the case for fractal or mul­ti­fractal sig­nals, as was proved by Alain Arneodo, Ur­i­el Frisch, Gior­gio Par­isi and their col­lab­or­at­ors. Stéphane Jaf­fard proved that the choice of the ana­lyz­ing wave­let was not very im­port­ant as long as \( \psi \) is suf­fi­ciently smooth, well loc­al­ized and has enough van­ish­ing mo­ments. The wave­let \( \psi(t)=\cos(5t)\exp(-t^2/2) \) (Mor­let’s fa­vor­ite) does not meet this third con­di­tion. The in­teg­ral of \( \psi \) is ex­tremely small but does not van­ish.

The con­tinu­ous wave­let ana­lys­is of a sig­nal \( s(t) \) is highly re­dund­ant. For some ana­lyz­ing wave­lets \( \psi \) the wave­let trans­form \( W(x, a) \) of a sig­nal \( s(t) \) is the solu­tion to a par­tial dif­fer­en­tial equa­tion. The ana­lys­is of a sig­nal of length \( N \) pro­duces \( N^2 \) coef­fi­cients. In some cases this re­dund­ancy can be use­ful and in oth­er cases the com­pu­ta­tion­al load is pro­hib­it­ive. At the oth­er ex­treme of the pic­ture we find the Fast Wave­let Trans­form which is de­scribed be­low.

The story I am telling now began on Janu­ary 15, 1985, when I met Alex Gross­mann in Mar­seilles for the first time. A month earli­er Jean Lascoux, a phys­i­cist and a col­league at Ecole Poly­tech­nique, had giv­en me a fas­cin­at­ing pre­print by Gross­mann and Mor­let. This pre­print was so at­tract­ive that I could not res­ist trav­el­ing to Mar­seilles. I spent three days talk­ing with Alex and I soon be­came his dis­ciple. I shared his val­ues and eth­ics. Among these val­ues I would single out an in­tense curi­os­ity, a great hu­mil­ity, a pro­found con­fid­ence in oth­ers and an out­stand­ing ca­pa­city for friend­ship. Alex be­came a spir­itu­al fath­er and a sci­entif­ic guide.

In a joint work with Alex and In­grid we con­struc­ted a frame of \( L^2({\mathbb R}^n) \) of the form \[ \psi_{j, k}(x)=2^{jn/2}\psi(2^jx-k),\quad j\in {\mathbb Z},\,k\in {\mathbb Z}^n, \] where the ana­lyz­ing wave­let \( \psi \) be­longs to the Schwartz class. The wave­let coef­fi­cients of a sig­nal \( s(t) \) are \( c(j,k)=\langle s, \psi_{j,k}\rangle. \) These coef­fi­cients are still re­dund­ant but the re­con­struc­tion of the sig­nal \( s(t) \) is ex­act. This \( \psi \) is named the moth­er wave­let since it gen­er­ates the oth­er wave­lets in the frame. Our goal was to build a di­git­al ver­sion of wave­let ana­lys­is which would be ex­act. Our work was en­titled “Pain­less nonortho­gon­al ex­pan­sions” and was pub­lished in J. Math. Phys. 27 (1986), 1271–1283. I had not yet met In­grid at that time. This ex­plains why in [1] In­grid is eager to meet.

However I was dis­sat­is­fied with this res­ult and I wondered if, in­stead of a frame, I could con­struct a true or­thonor­mal basis with the same struc­ture. Dur­ing the Sum­mer of 1985 I did it and I mailed the manuscript to In­grid. In the Fall of 1985 I gave a lec­ture at the Cour­ant In­sti­tute on this con­struc­tion. On Decem­ber 3, 1985, In­grid ac­know­ledged re­ceipt of my draft (let­ter [1] and apo­lo­gized for miss­ing my talk. My con­struc­tion was pub­lished six months later as “Prin­cipe d’in­cer­ti­tude, bases hil­ber­tiennes et algèbres d’opérat­eurs” (The un­cer­tainty prin­ciple, Hil­bert base and op­er­at­or al­geb­ras) in Sémin­aire Bourbaki, 38-ème année, Feb­ru­ary 1986, vol. 1985/86). Us­ing this basis every Schwartz dis­tri­bu­tion \( S \) is en­coded as a simple se­quence \( c_k,\,k=0, 1, \dots \) of num­bers and the in­tric­ate prop­er­ties of \( S \) be­come simple growth con­di­tions on this se­quence \( c_k. \) For a func­tion­al ana­lyst this was para­dise. However such wave­lets had in­fin­ite sup­ports and were use­less in real life prob­lems. One year later In­grid over­came that obstacle and con­struc­ted or­thonor­mal wave­let bases which are smooth (\( m \)-con­tinu­ous de­riv­at­ives) and com­pactly sup­por­ted. The moth­er wave­let de­pends on the re­quired reg­u­lar­ity \( m \) [6]. When com­pared with the deep­ness of In­grid’s achieve­ment my con­struc­tion looks trivi­al. In her let­ter of Ju­ly, 1986, [2] In­grid wanted to know wheth­er my con­struc­tion of an or­tho­gon­al wave­let basis could be mod­i­fied in such a way that only fine scales would play a role. She pro­posed a way for do­ing it. In a joint work with P-G. Lemarié I had already met this goal. This was pub­lished as “Ondelettes et bases hil­ber­tiennes,” Rev. Mat. Iberoam. 2:1-2, (1987), 1–18.

In Novem­ber 1986 I was in­vited to give a talk at the Uni­versity of Chica­go. It was the time when Stéphane Mal­lat was writ­ing his Ph.D. at the De­part­ment of Com­puter and In­form­a­tion Sci­ence of the Uni­versity of Pennsylvania. Stéphane was eager to talk with me and we met at Eck­hart Hall. Stéphane went there with some break­ing news. He had dis­covered that the con­struc­tion of or­thonor­mal wave­let bases was obey­ing the same al­geb­ra­ic rules as the design of quad­rat­ure mir­ror fil­ters. These fil­ters were already a sub­ject of in­tense in­vest­ig­a­tion in elec­tric­al en­gin­eer­ing. Stéphane was bridging the gap between math­em­at­ics and elec­tric­al en­gin­eer­ing. We had three days of in­tense work in Pro­fess­or Zyg­mund’s of­fice. Ant­oni Zyg­mund was still alive but let us freely use his of­fice. My main con­tri­bu­tion to this dis­cus­sion was to warn Stéphane that the it­er­at­ive pro­ced­ure yield­ing wave­lets could di­verge when ap­plied to some “bad” quad­rat­ure mir­ror fil­ters (see D. Esteban and C. Ga­land, “Ap­plic­a­tion of quad­rat­ure mir­ror fil­ters to split band voice cod­ing schemes,” Proc. IEEE ICAS­SP (1977), 191–195). This point was go­ing to be fully un­der­stood a few years later by Al­bert Co­hen and In­grid. Moreover we were un­able to char­ac­ter­ize the “good” quad­rat­ure mir­ror fil­ters that yield smooth wave­lets after it­er­a­tion. A few months later In­grid solved these two prob­lems and achieved her splen­did con­struc­tion of or­thonor­mal bases of com­pactly sup­por­ted smooth wave­lets ([4] and [5] writ­ten two days later). In [4] and [5] In­grid ex­plains her con­struct­ive vis­ion of com­pactly sup­por­ted scal­ing func­tions and wave­lets. This vis­ion was con­sist­ent with the frame­work in­tro­duced by Mal­lat and me, while bring­ing com­pletely nov­el in­sight to the needed al­geb­ra­ic ma­nip­u­la­tions to ob­tain the fil­ter banks that en­sure both com­pact sup­port and smooth­ness on the res­ult­ing wave­lets. A month later In­grid sent me the mag­ni­fi­cent let­ter [6] where she de­scribes her con­struc­tion of or­thonor­mal wave­let bases with com­pact sup­port.

As in­dic­ated in her let­ter [4] In­grid was con­cerned with a prob­lem raised by sci­ent­ists work­ing in com­puter vis­ion. They wanted some even (or odd) wave­lets while In­grid had proved that the Haar sys­tem is the only com­pactly sup­por­ted or­thonor­mal wave­let basis with this prop­erty. That is why In­grid and Al­bert Co­hen con­struc­ted smooth and sym­met­ric­al biortho­gon­al wave­lets (the ones which are used in the com­pres­sion stand­ard JPEG2000). I was eager to pub­lish their beau­ti­ful pa­per in La Rev­ista Matemática Iberoamer­ic­ana and In­grid sent me three cop­ies for sub­mis­sion to this journ­al. It was a time when e-mail did not ex­ist and a pa­per ver­sion of an art­icle was still needed to sub­mit it. In [8] (dated Decem­ber 1, 1987) In­grid says how much she is sorry for the death of my moth­er. Let me say today how much I was so moved by your kind let­ter, dear In­grid. In [9] (Janu­ary 15, 1990) In­grid ac­know­ledges re­ceipt of my book and be­gins dis­cuss­ing time-fre­quency wave­lets.

The pan­or­ama dra­mat­ic­ally changed around 1990. Mo­tiv­ated by a prob­lem raised by Ken­neth Wilson, In­grid, in a joint work with Stéphane Jaf­fard and Jean-Lin Journé, con­struc­ted an or­thonor­mal basis of time-fre­quency wave­lets. The prob­lem solved by In­grid et al. has a long his­tory. It was raised by Den­nis Gabor in 1946. Gabor thought that the func­tions \[ \exp(2\pi i k x)\exp(-(x-l)^2/2),\quad k,l\in{\mathbb{Z}}, \] could be a basis of \( L^2({\mathbb{R}}) \). When he raised this is­sue Gabor an­ti­cip­ated the di­git­al re­volu­tion. He tried to find an ef­fi­cient way to en­code a speech sig­nal. He be­lieved that a speech sig­nal \( s(x) \) could be en­coded as a short se­quence of num­bers \( c(k, l),\,k,l\in{\mathbb Z} \). These num­bers would have been the coef­fi­cients of \( s(x) \) in the Gabor basis. Un­for­tu­nately Gabor was wrong and the Gabor basis is not a basis. Something was miss­ing. It is like omit­ting the num­ber 7 in arith­met­ic. In­grid could fix the prob­lem and her dis­cov­ery (a joint work with Jaf­fard and Journé) happened to be sem­in­al in the de­tec­tion of grav­it­a­tion­al waves (see Sergey Kli­men­ko et al., “Ob­serving grav­it­a­tion­al-wave tran­si­ent GW150914 with min­im­al as­sump­tions” at ht­tps://dcc.ligo.org/LIGO-P1500229/pub­lic/main).

In [11] In­grid is in­ter­ested in the ap­plic­a­tion of wave­lets to sci­entif­ic com­put­ing. She dis­cusses a new al­gorithm dis­covered by Greg Beylkin, Ron­ald Coi­f­man and Vladi­mir Rokh­lin. The goal is to ac­cel­er­ate the com­pu­ta­tion of the product of two large \( N\times N \) matrices un­der the hy­po­thes­is that these matrices are al­most di­ag­on­al in a wave­let basis. The charm­ing let­ters [12] and [13] of­fer a mov­ing pic­ture of a preg­nant In­grid.

The let­ter [16] is not dated but I would say that it was writ­ten circa 1992 since In­grid’s daugh­ter Car­oline be­gins to walk. Soon after the con­struc­tion by In­grid et al. of or­thonor­mal bases of time-fre­quency wave­lets Coi­f­man and I found an­oth­er solu­tion in which the time seg­ment­a­tion could be ar­bit­rar­ily im­posed. In [16] In­grid with her usu­al fair­ness stresses that we should men­tion the con­tri­bu­tions of H. Mal­var, J. P. Prin­cen, and A. B. Brad­ley who an­ti­cip­ated our con­struc­tion un­der the name of lapped trans­forms. And Mar­tin Vet­terli should not be for­got­ten! Or­thonor­mal bases of time-scale wave­lets pre­vi­ously ex­is­ted un­der the name of quad­rat­ure mir­ror fil­ters, as Mal­lat dis­covered. Sim­il­arly or­thonor­mal bases of time-fre­quency wave­lets already ex­is­ted as lapped trans­forms. Ana­lysts should be mod­est. But in [16] In­grid is mostly dis­cuss­ing the con­struc­tion of an or­thonor­mal wave­let basis on a giv­en in­ter­val. Everything needs to be loc­al­ized on this spe­cif­ic in­ter­val, which was not the case for the pre­ced­ing con­struc­tions. This paves the way to the prob­lems dis­cussed in the let­ter [17].

The let­ter [17] is ex­tremely in­ter­est­ing and ex­em­pli­fies the dif­fer­ences between In­grid and me. In­grid is at the same time a math­em­atician, a phys­i­cist and a sci­ent­ist. She im­me­di­ately per­ceived the tal­ent of Wim Sweldens. In­grid wished that Wim would be hired by Prin­ceton. Prin­ceton asked for my ad­vice. Be­fore giv­ing my opin­ion I wrote sev­er­al emails to In­grid. In­grid fi­nally answered but her mail broke down while she was writ­ing. This was for­tu­nate and we have this long and beau­ti­ful let­ter. In­grid dis­cusses the rel­ev­ance of math­em­at­ics in pro­gram­ming and states that ef­fi­cient al­gorithms shall be based on good and deep math­em­at­ics. This is a mar­velous state­ment. Fi­nally Wim moved to high tech and is fam­ous for sev­er­al spec­tac­u­lar in­nov­a­tions.

In­grid’s let­ters of­fer a genu­ine in­sight in­to this ex­cep­tion­al dec­ade in which wave­let ana­lys­is was elab­or­ated. In these let­ters In­grid is act­ively present with her sci­entif­ic vis­ion, her en­thu­si­asm, her doubts, her fair­ness, and her ex­traordin­ary abil­ity to com­mu­nic­ate. In­grid trus­ted me and nev­er hes­it­ated to un­veil her deep­est thoughts and feel­ings. Thanks, In­grid